2 Answers
2

The previously accepted answer is incorrect at least with regard to the Queen's British citizenship. Although the reigning monarch is arguably not a British subject, the current Queen is a British citizen according to law. These are distinct legal terms, regardless of what the OED says. The term 'British subject' is currently very rarely used, and until 1983 was mostly an unofficial term. Even after 1983, it is only applied in fairly niche cases.

However, the term British citizen is far more regularly used. Section 11(1) of the British Nationality Act 1981, which defined the term, states:

11 Citizens of U.K. and Colonies who are to become British citizens at
commencement.

(1) Subject to subsection (2), a person who immediately
before commencement—

(a) was a citizen of the United Kingdom and
Colonies; and

(b) had the right of abode in the United Kingdom under
the Immigration Act 1971 as then in force,shall at commencement
become a British citizen.

(subsection (2) is not relevant in this case)

So we need to go back further, to check whether the Queen was a citizen of the United Kingdom and Colonies on January 1st 1983. This term was introduced by the British Nationality Act 1948.

Part II Section 4 of this Act states:

4 Citizenship by birth

Subject to the provisions of this section, every person born within the United Kingdom and Colonies after the
commencement of this Act shall be a citizen of the United Kingdom and
Colonies by birth:

Provided that a person shall not be such a citizen
by virtue of this section if at the time of his birth—

(a) his father possesses such immunity from suit and legal process as is accorded to an envoy of a foreign sovereign power accredited to
His Majesty, and is not a citizen of the United Kingdom and Colonies;
or

(b) his father is an enemy alien and the birth occurs in a place then under occupation by the enemy.

Neither of these exemptions apply to the Queen, and so by Section 12 of the Act:

12 British subjects before commencement of Act becoming citizens of
United Kingdom and Colonies

(1) A person who was a British subject
immediately before the date of the commencement of this Act shall on
that date become a citizen of the United Kingdom and Colonies if he
possesses any of the following qualifications, that is to say—

(a) that
he was born within the territories comprised at the commencement of
this Act in the United Kingdom and Colonies, and would have been such
a citizen if section four of this Act had been in force at the time of
his birth;

(b) that he is a person naturalised in the United Kingdom
and Colonies;

(c) that he became a British subject by reason of the
annexation of any territory included at the commencement of this Act
in the United Kingdom and Colonies.

The Queen was born in London, and as we have determined that section 4 applies to her, she became a citizen of the United Kindom and Colonies on the date of commencement of the Act. Note that this applies only if she was a British subject at that point, which is of course now a bone of contention. However, as she did not ascend the throne until 1952, she was definitely a British subject at that point; a subject of her father, George VI. At that point, the term British subject applied to anyone in the British Empire and Commonwealth.

So in summary, as she was a British subject in 1948, this means she was a citizen of the United Kingdom and Colonies in 1983, which means she is a British citizen today. Whether she is a British subject is fairly irrelevant, as her status as a British citizen gives her all the rights and more, and the legal term is barely used nowadays. Although it may seem a bit odd that the Queen can be a citizen of a country which she rules, there is no law or exemption which excludes the Monarch from any of the above Acts, nor any other legislation which changes her citizenship status. This is probably because, in the grand scheme of things, it really doesn't matter - the Queen is hardly likely to be deported or refused consular assistance, and she doesn't need a passport.

Excellent research that actually looks into the legal status instead of popular musings. The only thing I could see that might be an issue for the Queen is that she might not be a "person" under the laws upon becoming the Monarch. But to my knowledge, she is still a person and the Crown immunity only excludes laws that put limits on things, not ones that give rights (e.g. citizenship).
– zhantongzApr 17 at 12:36

It would be odd to describe the Queen as ruler and subject to herself. As to the Royal Family, they are subjects of the Queen, but retain citizenship of the country of origin, unless naturalized

It was not unusual to naturalize husbands (and also sometimes wives) of members of the British Royal Family. [...] 49 & 48 Vict c.1 (1885): private act naturalizing Prince Henry of Battenberg future husband of Princess Beatrice, daughter of Queen Victoria.